Let $\mathfrak{g}$ be a complex semisimple Lie algebra, let $\mathfrak{h} \subset \mathfrak{g}$ be a Cartan subalgebra, and let $\Phi \subset \mathfrak{h}^*$ be the root system of $\mathfrak{g}$ relative to $\mathfrak{h}$. For each $\gamma \in \mathfrak{h}^*$, define
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\begin{align*}
\mathfrak{g}_\gamma
=
\{z \in \mathfrak{g} : [h,z] = \gamma(h)z \text{ for every } h \in \mathfrak{h}\},
\end{align*}
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with the convention that $\mathfrak{g}_\gamma = 0$ when $\gamma \notin \Phi \cup \{0\}$. Then, for every $\alpha,\beta \in \mathfrak{h}^*$,